Calculate (1/5)³: Solving the Cube of a Simple Fraction

Q: Why do I need to apply the exponent to both parts of the fraction?

The power rule for fractions states that $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $. When you raise a fraction to a power, you're multiplying the entire fraction by itself that many times, so both numerator and denominator get the exponent!

Q: What's the difference between $ \frac{1^3}{5^3} $ and $ \frac{1^3}{5} $ ?

$ \frac{1^3}{5^3} = \frac{1}{125} $ is correct because both parts get cubed. $ \frac{1^3}{5} = \frac{1}{5} $ is wrong because the denominator wasn't cubed. Always apply the exponent to both numerator and denominator!

Q: Can I just multiply $ \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} $ instead?

Yes! That's exactly what the power rule does. $ \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} = \frac{1 \times 1 \times 1}{5 \times 5 \times 5} = \frac{1^3}{5^3} $. The power rule is just a shortcut for repeated multiplication!

Q: Why does $ 1^3 $ still equal 1?

Any power of 1 always equals 1! So $ 1^3 = 1 \times 1 \times 1 = 1 $. This is why $ \left(\frac{1}{5}\right)^3 = \frac{1^3}{5^3} = \frac{1}{5^3} = \frac{1}{125} $.

Q: How do I know when to leave the answer as $ \frac{1^3}{5^3} $ versus $ \frac{1}{125} $ ?

Check what the question asks for! If it wants the expression form (showing the work), leave it as $ \frac{1^3}{5^3} $. If it wants the simplified numerical answer, calculate to get $ \frac{1}{125} $.

Fraction Exponents with Power Rules

Insert the corresponding expression:

$\left(\frac{1}{5}\right)^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the laws of exponents, a fraction raised to a power (N)

00:07 equals the numerator and denominator raised to the same power (N)

00:11 We will apply this formula to our exercise

00:15 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{1}{5}\right)^3=$

Step-by-step solution

To solve this problem, we need to simplify the expression $\left(\frac{1}{5}\right)^3$ using the exponent rules for fractions:

Step 1: Identify the base and the exponent. Here, the base is $\frac{1}{5}$ and the exponent is $3$ .
Step 2: Apply the rule for raising a fraction to a power. The rule states that $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .
Step 3: Apply the exponent to both the numerator and the denominator:

Thus, $\left(\frac{1}{5}\right)^3 = \frac{1^3}{5^3}$ .

Therefore, the simplified expression is $\frac{1^3}{5^3}$ , which corresponds to choice 1 in the provided options.

Final Answer

$\frac{1^3}{5^3}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply exponent to both numerator and denominator separately
Technique: $\left(\frac{1}{5}\right)^3 = \frac{1^3}{5^3}$ keeps fraction structure
Check: Verify $\frac{1^3}{5^3} = \frac{1}{125}$ using basic exponent rules ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only the denominator
Don't write $\left(\frac{1}{5}\right)^3 = \frac{1}{5^3}$ = incorrect result! This ignores the power rule for fractions and gives the wrong expression. Always apply the exponent to both the numerator AND denominator using $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do I need to apply the exponent to both parts of the fraction?

The power rule for fractions states that $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ . When you raise a fraction to a power, you're multiplying the entire fraction by itself that many times, so both numerator and denominator get the exponent!

What's the difference between $\frac{1^3}{5^3}$ and $\frac{1^3}{5}$ ?

$\frac{1^3}{5^3} = \frac{1}{125}$ is correct because both parts get cubed. $\frac{1^3}{5} = \frac{1}{5}$ is wrong because the denominator wasn't cubed. Always apply the exponent to both numerator and denominator!

Can I just multiply $\frac{1}{5} \times \frac{1}{5} \times \frac{1}{5}$ instead?

Yes! That's exactly what the power rule does. $\frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} = \frac{1 \times 1 \times 1}{5 \times 5 \times 5} = \frac{1^3}{5^3}$ . The power rule is just a shortcut for repeated multiplication!

Why does $1^3$ still equal 1?

Any power of 1 always equals 1! So $1^3 = 1 \times 1 \times 1 = 1$ . This is why $\left(\frac{1}{5}\right)^3 = \frac{1^3}{5^3} = \frac{1}{5^3} = \frac{1}{125}$ .

How do I know when to leave the answer as $\frac{1^3}{5^3}$ versus $\frac{1}{125}$ ?

Check what the question asks for! If it wants the expression form (showing the work), leave it as $\frac{1^3}{5^3}$ . If it wants the simplified numerical answer, calculate to get $\frac{1}{125}$ .

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Calculate (1/5)³: Solving the Cube of a Simple Fraction

Fraction Exponents with Power Rules

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to apply the exponent to both parts of the fraction?

What's the difference between $\frac{1^3}{5^3}$ and $\frac{1^3}{5}$ ?

Can I just multiply $\frac{1}{5} \times \frac{1}{5} \times \frac{1}{5}$ instead?

Why does $1^3$ still equal 1?

How do I know when to leave the answer as $\frac{1^3}{5^3}$ versus $\frac{1}{125}$ ?

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More Questions

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Calculate (6/8)² : Square of a Fraction Problem

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Calculate (4/7)³: Evaluating the Cube of a Fraction

Calculate (1/5)³: Solving the Cube of a Simple Fraction

Fraction Exponents with Power Rules

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to apply the exponent to both parts of the fraction?

What's the difference between 1353 \frac{1^3}{5^3} 5313​ and 135 \frac{1^3}{5} 513​?

Can I just multiply 15×15×15 \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} 51​×51​×51​ instead?

Why does 13 1^3 13 still equal 1?

How do I know when to leave the answer as 1353 \frac{1^3}{5^3} 5313​ versus 1125 \frac{1}{125} 1251​?

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More Questions

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Calculate (6/8)² : Square of a Fraction Problem

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Calculate (2/3)²: Finding the Square of a Fraction

Calculate (4/7)³: Evaluating the Cube of a Fraction

What's the difference between $\frac{1^3}{5^3}$ and $\frac{1^3}{5}$ ?

Can I just multiply $\frac{1}{5} \times \frac{1}{5} \times \frac{1}{5}$ instead?

Why does $1^3$ still equal 1?

How do I know when to leave the answer as $\frac{1^3}{5^3}$ versus $\frac{1}{125}$ ?